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Senary Numbers (Base 6) | Examples, questions and answers

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Senary Numbers

In science and computer science, the system of the senary numbers are a number system in which numbers are represented using the digits from 0 to 5 (0-5). It can be used as a checking tool, along with the octal system and the sexagesimal system. The senary system or sexagesimal system is based on a number (x).

Exponentiation

Senary numbers use only six digits, digits increase faster than other bases. But, the exponents of two and three are equal, this can be expressed as: 10(6)n = 2(6)n × 3(6)n.

In particular, six and ten have the same structure whose prime factors have the same exponents. Also, six to the 4n-th power (104n) is close to ten to the 3n-th power (143n). For instance:

Factorization of basic prime numbers:
Power of six by senary notation
The power of six by senary notation
Exponent Senary Decimal equivalent Duodecimal equivalent Vigesimal equivalent
1 10 6 6 6
2 100 62 = 36 62 = 30 62 = 1G
3 1 000 63 = 216 63 = 160 63 = AG
4 10 000 64 = 1 296 64 = 900 64 = 34G
5 100 000 65 = 7 776 65 = 4 600 65 = J8G
10 1 000 000 66 = 46 656 66 = 23 000 66 = 5 GCG
11 10 000 000 67 = 279 936 67 = 116 000 67 = 1E JGG
12 100 000 000 68 = 1 679 616 68 = 690 000 68 = A9 J0G
13 1 000 000 000 69 = 10 077 696 69 = 3 460 000 69 = 32J E4G
14 10 000 000 000 610 = 60 466 176 6A = 18 300 000 6A = IHI 58G
15 100 000 000 000 611 = 362 797 056 6B = A1 600 000 6B = 5 D79 CCG
20 1 000 000 000 000 612 = 2 176 782 336 610 = 509 000 000 6C = 1E 04H FGG
21 10 000 000 000 000 613 = 13 060 694 016 611 = 2 646 000 000 6D = A4 196 F0G
22 100 000 000 000 000 614 = 78 364 164 096 612 = 13 230 000 000 6E = 314 8G0 A4G
23 1 000 000 000 000 000 615 = 470 184 984 576 613 = 77 160 000 000 6F = I76 CG3 18G
24 10 000 000 000 000 000 616 = 2 821 109 907 456 614 = 396 900 000 000 6G = 5 A3J GGI 8CG
25 100 000 000 000 000 000 617 = 16 926 659 444 736 615 = 1 A94 600 000 000 6H = 1D 13J 11A BGG
30 1 000 000 000 000 000 000 618 = 101 559 956 668 416 616 = B 483 000 000 000 6I = 9I 73E 693 B0G

Integer notation (whole number that can be positive, negative, or zero)

From senary to decimal

Here are the first numbers from 1 to 40 and from 91 to 110 expressed in senary and then decimal positional notation.

Senary 1 2 3 4 5 10 11 12 13 14 15 20 21 22 23 24 25 30 31 32
Decimal 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Senary 33 34 35 40 41 42 43 44 45 50 51 52 53 54 55 100 101 102 103 104
Decimal 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Senary 231 232 233 234 235 240 241 242 243 244 245 250 251 252 253 254 255 300 301 302
Decimal 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

Senary numbers are expresses six as “10”, nine (9) as “13” i.e. “six plus three”, ten (decimal 10) as “14” i.e. “six plus four”, twelve (decimal 12) as “20” i.e. “two sixes”, sixteen (decimal 16) as “24” i.e. “two sixes and four“.

Digits for multiples of three end in 3 or 0, e.g. decimal 18 (eighteen) is expressed as “30” (three sixes), decimal 15 (ten and five) is expressed as “23” (two sixes and three), decimal 27 (two ten and seven) is expressed as “43” (four sixes and three).

Numbers over 100 (decimal 36), e.g. decimal 81 is expressed as “213” to say “two of six squares, six and three”, decimal 100 is expressed as “244” to say “two of six squares, four six and four“.

Notation rating breakdown:
Examples of arithmetic operations
Examples of arithmetic operations
Decimal Senary
1944 + 56 = 2000 13000 + 132 = 13132
100 – 64 = 36 244 – 144 = 100
16 × 81 = 1296 24 × 213 = 10000
1080 ÷ 27 = 40 5000 ÷ 43 = 104
64 / 144 = 4 / 9 144 / 400 = 4 / 13
38 = 6561 312 = 50213
24 = 23×3 40 = 23×3
Date and hour
Date and hour
Events Decimal Senary
The death of Alfred Nobel 10 / 12 / 1896 14 / 20 / 12440
Atomic bombing of Hiroshima 6 / 8 / 1945 10 / 12 / 13001
September 11, 2001 attacks 11 / 9 / 2001 15 / 13 / 13133
From decimal to senary

Here are some benchmarks.

Decimal 1 2 3 4 5 6 7 8 9 12 15 18 24 27 30 36
Senary 1 2 3 4 5 10 11 12 13 20 23 30 40 43 50 100
Decimal 42 54 72 108 144 162 180 216 324 432 648 972 1080 1296 1944 2592
Senary 110 130 200 300 400 430 500 1000 1300 2000 3000 4300 5000 10000 13000 20000

As will be described in detail in the section on fractions, the upper-digit shift of senary numbers has a relationship of “four to nine” (4×13 = 100).

Therefore, four 130s will be 1000, nine 400s will be 10000, three quarters of 1000 will be 430, two ninths of 100 will be 12.

Doubles or duplicate detection

(as in decimal, all numbers ending in a digit representing a multiple of 2 — i.e. 2, 4, 6, 8, 0 are divisible by 2; and all numbers ending in a multiple of 5 — i.e. 5 and 0 — divisible by 5).

Prime number

A prime number other than 2 or 3 can therefore only end in senary with 1 or 5. (in decimal a prime number other than 2 or 5 can only end with 1, 3, 7 or 9).

Fractions and divisions

Six is the product of two prime numbers, namely 2 and 3. As a result, some properties of senary positional notation are reminiscent of those of decimal positional notation.

All fractions whose denominator knows no other prime factor than 2 and 3 are expressed in senary with a finite number of decimal places. (Compare with the role of 2 and 5 in decimal.) Six and ten are only even numbers, a quarter is expressed as two decimal places. Thus, senary and decimal system, the position of 3 and 5 is reversed. For example, “0.2” is 1/5 (i.e. two tenths) in decimal, but 1/3 (i.e. two sixths) in senary.

In senary notation, reciprocals of powers of 2 are powers of 3, reciprocals of powers of 3 are powers of 2, dividing by powers of 2 and 3 becomes easier than any notation. Thus, the powers of 3 become dominant, the powers of 5 become weak.

The senary fraction have the characteristic of “short repetition” as being the same as the decimal fraction. Decimal fraction have 3-3 requires 3 digit repeats, 3-4 requires 9 (=3-2) digit repeats. Like this, the senary fraction have 5-2 requires 5-digit repetitions. The number whose repetitions reach about twenty-seven is 3-5 in decimal (33, twenty-seven digits), 5-3 in senary (5², twenty-five digits).

Unit fraction

Factorisation Décimal Sénaire
2 1/2 = 0,5 1/2 = 0,3
3 1/3 = 0,33 repetition 1/3 = 0,2
22 1/4 = 0,25 1/4 = 0,13
5 1/5 = 0,2 1/5 = 0,11 repetition
2×3 1/6 = 0,166 répétition 1/10 = 0,1
11 1/7 = 0,142857142857 repetition 1/11 = 0,0505 repetition
23 1/8 = 0,125 1/12 = 0,043
32 1/9 = 0,11 repetition 1/13 = 0,04
2×5 1/10 = 0,1 1/14 = 0,033 repetition
15 1/11 = 0,0909 repetition 1/15 = 0,03134524210313452421 repetition
22×3 1/12 = 0,08333 repetition 1/20 = 0,03
21 1/13 = 0,076923076923 repetition 1/21 = 0,024340531215024340531215 repetition
2×11 1/14 = 0,0714285714285 repetition 1/22 = 0,02323 repetition
3×5 1/15 = 0,066 repetition 1/23 = 0,022 repetition
24 1/16 = 0,0625 1/24 = 0,0213
2×32 1/18 = 0,055 repetition 1/30 = 0,02
22×5 1/20 = 0,05 1/32 = 0,0144 repetition
23×3 1/24 = 0,04166 repetition 1/40 = 0,013
52 1/25 = 0,04 1/41 = 0.0123501235 repetition
33 1/27 = 0,037037 repetition 1/43 = 0,012
25 1/32 = 0,03125 1/52 = 0,01043
22×32 1/36 = 0,0277 repetition 1/100 = 0,01
23×5 1/40 = 0,025 1/104 = 0,00522 repetition
24×3 1/48 = 0,020833 repetition 1/120 = 0,0043
2×52 1/50 = 0,02 1/122 = 0,00415304153 repetition
2×33 1/54 = 0,0185185 repetition 1/130 = 0,004
210 1/64 = 0,015625 1/144 = 0,003213
23×32 1/72 = 0,01388 répétition 1/200 = 0,003
24×5 1/80 = 0,0125 1/212 = 0,002411 repetition
34 1/81 = 0,012345679012345679 répétition 1/213 = 0,0024
25×3 1/96 = 0,0104166 repetition 1/240 = 0,00213
22×52 1/100 = 0,01 1/244 = 0,002054320543 repetition
22×33 1/108 = 0,00925925 repetition 1/300 = 0,002
53 1/125 = 0,008 1/325 = 0,0014211153224043351545031
0014211153224043351545031 repetition
211 1/128 = 0,0078125 1/332 = 0,0014043
24×32 1/144 = 0,006944 repetition 1/400 = 0,0013
25×5 1/160 = 0,00625 1/424 = 0,0012033 repetition
2×34 1/162 = 0,0061728395061728395 repetition 1/430 = 0,0012
210×3 1/192 = 0,00520833 repetition 1/520 = 0,001043
23×52 1/200 = 0,005 1/532 = 0,0010251402514 repetition
23×33 1/216 = 0,004629629 repetition 1/1000 = 0,001

Main fraction

Enter the decimal equivalent in parentheses.

Up to ninths (except sevenths and eighths)

  • 1/2 = 0,3
  • 1/3 = 0,2
  • 2/3 = 0,4
  • 1/4 = 0,13 (9/36)
  • 3/4 = 0,43 (27/36)
  • 1/5 = 0.1111…
  • 2/5 = 0.2222…
  • 3/5 = 0.3333…
  • 4/5 = 0.4444…
  • (1/6)dix = 1/10 = 0,1
  • (5/6)dix = 5/10 = 0,5
  • (1/9)dix = 1/13 = 0,04 (4/36)
  • (2/9)dix = 2/13 = 0,12 (8/36)
  • (4/9)dix = 4/13 = 0,24 (16/36)
  • (5/9)dix = 5/13 = 0,32 (20/36)
  • (7/9)dix = 11/13 = 0,44 (28/36)
  • (8/9)dix = 12/13 = 0,52 (32/36) (approximation of decimal 0.9)
Tenths
Twelfths (1 / 22×3)
Eighteenth (1 / 2×32)
Eighths (2-3)
Twenty-sevenths  (3-3)
  • (1/27)ten = 1/43 = 0,012 (8/216)
  • (2/27)ten = 2/43 = 0,024 (16/216)
  • (4/27)ten = 4/43 = 0,052 (32/216)
  • (5/27)ten = 5/43 = 0,104 (40/216)
  • (7/27)ten = 11/43 = 0,132 (56/216)
  • (8/27)ten = 12/43 = 0,144 (64/216)
    (decimal approximation 0,3)
  • (10/27)ten = 14/43 = 0,212 (80/216)
  • (11/27)ten = 15/43 = 0,224 (88/216)
    (decimal approximation 0,4)
  • (13/27)ten = 21/43 = 0,252 (104/216)
  • (14/27)ten = 22/43 = 0,304 (112/216)
  • (16/27)ten = 24/43 = 0,332 (128/216)
    (decimal approximation 0,6)
  • (17/27)ten = 25/43 = 0,344 (136/216)
  • (19/27)ten = 31/43 = 0,412 (152/216)
    (decimal approximation 0,7)
  • (20/27)ten = 32/43 = 0,424 (160/216)
  • (22/27)ten = 34/43 = 0,452 (176/216)
  • (23/27)ten = 35/43 = 0,504 (184/216)
  • (25/27)ten = 41/43 = 0,532 (200/216)
  • (26/27)ten = 42/43 = 0,544 (208/216)

Calculation examples

1/2, 1/4
1 / 23 (1/8 in decimal)
1/3
1/9, 1/100 in senary (1/36 in decimal)
1 / 33 (1/27 in decimal), 1/1000 in senary (1/216 in decimal)
1/5
1 / 52 (1/25 in decimal), 1/100 in decimal
1 / 24 (1/16 in decimal)
1 / 34 (1/81 in decimal)

Sources: PinterPandai,

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